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Nilpotent subspaces of maximal dimension in semi-simple Lie algebras. (English) Zbl 1163.17011

The main theorem of the article states that any linear vector subspace \(V\) of a complex semi-simple Lie algebra \(\mathfrak{g}\) consisting of ad-nilpotent elements has dimension at most \(1/2(\dim\mathfrak{g}-\mathrm{rk}\,\mathfrak{g})\) and if equality holds then the vector space \(V\) is the nilradical of a Borel subalgebra of \(\mathfrak{g}\).
The first part of the statement has already been known [R. Meshulam and N. Radwan, Linear Algebra Appl. 279, No. 1-3, 195–199 (1998; Zbl 0941.17004)]. The authors provide an easy argument for it in Section 1. The difficulty lies in proving the second part of the theorem.
The authors prove a more general statement (Theorem 1): they consider the case of a connected reductive algebraic group \(G\) over an algebraically closed field \(K\), under a few additional assumptions on \(G\) (conditions 1, 2 and 3) that ensure that certain subgroups do not occur in \(G\). Let \(\mathfrak{g}\) be the Lie algebra of \(G\) and \(\mathfrak{n}\) be the Lie algebra of a maximal unipotent subgroup of \(G\).
Theorem 1 then states: If \(V\) is a nilpotent subspace of the Lie algebra \(\mathfrak{g}\), then \(\dim V\leq \dim\mathfrak{n}\) and if equality holds then \(V\) is conjugate to \(\mathfrak{n}\).
Section 1 contains the main results and an outline of the proof of Theorem 1. In Section 2, the dimension bound (for nilpotent subspaces) is established and \(T\)-stable nilpotent subspaces of this maximal dimension are studied (\(T\) a maximal torus of \(G\)). Section 3 presents the proof of Theorem 1.

MSC:

17B20 Simple, semisimple, reductive (super)algebras
20G05 Representation theory for linear algebraic groups

Citations:

Zbl 0941.17004
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